We can simplify the equation by finding a common denominator for all the fractions. The least common multiple of n, n+1, and n-1 is n(n+1)(n-1). Multiplying through the equation by n(n+1)(n-1), we get:
2(n+1)(n-1) + 2n(n-1) = r*2n(n-1) - n(n+1)
Simplifying the left side, we get:
2(n^2-1) + 2n^2(n-1) = r*2n(n-1) - n^2 - n
Expanding and collecting like terms on both sides, we get:
2n^2 - 2 + 2n^3 - 2n^2 = 2n^2r - n^2 - n
Simplifying further, we have:
2n^3 = 2n^2r - n
Dividing both sides by 2n, we get:
n^2 = nr - 1/2
Rearranging terms, we have:
nr = n^2 + 1/2
Therefore, the expression for r in terms of n is:
r = (n^2 + 1/2) / n
For a positive integer n, the rational number r satisfies
2n/n + 2n/(n + 1) = r*2n/n - n/(n - 1)
Express r in terms of n. (Your expression should be as simplified as possible.)
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